Obstructions to the Hasse principle for homogeneous spaces

The Hasse principle for homogeneous spaces under connected linear algebraic groups over number fields is well understood, with a number of deep theorems concerning principal homogeneous spaces under simply connected groups related to Serre’s Conjecture II. It is known that the Brauer-Manin obstruction is the only obstruction to the Hasse principle if the stabilizer of the homogeneous space is connected. We shall explain several parallel results and conjectures if the underlying field is the function field of a $p$-adic curve.